Hecke operators, buildings and Hall algebras

Serre's theory of trees has been applied successfully to calculations with automorphic forms for PGL(2) whenever strong approximation was sufficiently well working. This is, for instance, the case for rational function fields. In general, the class group becomes an obstruction, and a global variant of Serre's theory is needed.

In this talk, we introduce such a variant: the graph of a Hecke operator. We explain a structure theorem for elliptic function fields and its applications to automorphic forms. We investigate its connection to Ronan's theory of adelic buildings, which is work in progress with Robert Kremser. We line out how Burban and Schiffmann's result about the Hall algebra of an elliptic curve can be used to determine the graphs of Hecke operators for GL(n), which is a recent result by Roberto Alvarenga.